Finite Group Databasebeta

← #80990

Subgroups — #80990

Order 384.

374 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5732C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic
3C3×#474C211yesmaximal, characteristic, complemented
4#5078C211yesmaximal, characteristic, complemented
5C4×#214C211yesmaximal, characteristic, complemented
6#4884C212yesmaximal, complemented
7#4881C212yesmaximal, complemented
8#4851C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4780C212yesmaximal, complemented
11#4779C212yesmaximal, complemented
12#3353—31nomaximal, Sylow
13C2×#235—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16#947—21no—
17Q8×Dic3—21no—
18#938C2212yes—
19C2×#214C2211yescharacteristic
20#924—21no—
21C2×ΓL(1,𝔽25)C2212yes—
22#898—22no—
23#897C2212yes—
24#892C2212yes—
25D3×C4⋊C4C2212yes—
26#883—22no—
27C4×D12C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30C3×#107C2212yes—
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2211yescharacteristic
34(C2×D3)⋊C8C2212yes—
35#817C2212yes—
36#816C2212yes—
37#810C2211yescharacteristic
38#808C2211yescharacteristic
39#807C2212yes—
40#807C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#547—31no—
44#500—31no—
45#484—32no—
46#482—32no—
47#475—31no—
48#474D311yescharacteristic, complemented
49#468—32no—
50C4×SD16—31no—
51#436—31no—
52#428—32no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57#235—44no—
58C2×D12D411yescharacteristic
59C2×C4×D3—22no—
60C2×C4×D3C2312yes—
61C2×Dic6—21no—
62C2×Dic6C2311yescharacteristic
63C2×C3×C8C2312yes—
64C3×C4⋊C4—21no—
65C3×C4⋊C4C2312yes—
66C3×C4⋊C4C2311yescharacteristic
67C3×C22⋊C4—21no—
68C3×C42C2311yescharacteristic
69#217—22no—
70#214—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4D412yes—
74C4⋊Dic3—21no—
75C4⋊Dic3—22no—
76C4⋊Dic3D411yescharacteristic
77C4⋊Dic3C2312yes—
78C4⋊Dic3C2311yescharacteristic
79Dic3⋊C4—21no—
80Dic3⋊C4—22no—
81C4×Dic3—21no—
82C4×Dic3—22no—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85ΓL(1,𝔽25)—24no—
86C2×#41—61no—
87C2×SD16—31no—
88C2×M16—32no—
89C4⋊Q8—31no—
90C4⋊Q8C2×D311yescharacteristic
91#126—62no—
92#124—61no—
93#123—62no—
94C22⋊Q8—32no—
95C22⋊Q8—61no—
96#121—32no—
97C4×Q8—31no—
98C4×Q8—61no—
99C4×D4—31no—
100C4×D4C2×D311yescharacteristic
101#117—32no—
102C2×C4⋊C4—32no—
103#107—32no—
104#107—32no—
105#107C2×D312yes—
106#106—32no—
107#106—31no—
108C4⋊C8C2×D311yescharacteristic
109#103—32no—
110#103—31no—
111#102—31no—
112#102C2×D312yes—
113C22⋊C8—32no—
114C4×C8—31no—
115C23×C3—21no—
116C22×D3C2×D412yes—
117C3×Q8—42no—
118C3×D4—21no—
119C3×D4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—42no—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2411yescharacteristic
126#67—44no—
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3C2×D412yes—
131D12—22no—
132C4×D3—22no—
133C4×D3—22no—
134C4×D3—44no—
135Dic6—21no—
136Dic6—21no—
137Dic6—42no—
138C3×C8—22no—
139C3⋊C8—21no—
140C3⋊C8#4112yescomplemented
141#41—124no—
142C2×Q8—21no—
143C2×Q8—31no—
144C2×Q8—61no—
145C2×D4—31no—
146C2×D4—62no—
147C2×D4C22×D311yescharacteristic
148C22×C4—21no—
149C22×C4—32no—
150C22×C4—62no—
151SD16—62no—
152M16—64no—
153C2×C8—31no—
154C2×C8—31no—
155C2×C8C22×D312yes—
156C4⋊C4—21no—
157C4⋊C4—31no—
158C4⋊C4—31no—
159C4⋊C4—32no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4C22×D312yes—
165C4⋊C4C22×D311yescharacteristic
166C22⋊C4—21no—
167C22⋊C4—32no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C42—62no—
172C42—61no—
173C42C22×D311yescharacteristic
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—41no—
177C22×C3C22×D411yescharacteristic
178C2×D3—22no—
179C2×D3—22no—
180C2×D3—22no—
181C3×C4—21no—
182C3×C4—41no—
183C3×C4—42no—
184C3×C4AGL(1,ℤ/8)12yes—
185C3×C4C2×#4111yescharacteristic
186C3×C42−1+411yescharacteristic
187Dic3—22no—
188Dic3—41no—
189Dic3—42no—
190Dic3—41no—
191C23—21no—
192C23—32no—
193Q8—42no—
194Q8—61no—
195Q8—61no—
196Q8—122no—
197D4—21no—
198D4—21no—
199D4—62no—
200D4—124no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—32no—
204C2×C4—42no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—62no—
210C2×C4—124no—
211C2×C4D3×D411yescharacteristic
212C2×C4D3×D411yescharacteristic
213C2×C4C23×D311yescharacteristic
214C8—22no—
215C8—32no—
216C8—61no—
217C2×C3—81no—
218C2×C3#54711yescharacteristic
219C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C3#57811yescharacteristic
221D3—42no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—62no—
226C22—62no—
227C22—62no—
228C22C2×D3×D411yescharacteristic
229C4—21no—
230C4—41no—
231C4—42no—
232C4—62no—
233C4—122no—
234C4—121no—
235C4—121no—
236C4#91112yes—
237C4#100711yescharacteristic
238C4D3×#4111yescharacteristic
239C3#335311yescharacteristic, complemented, Sylow
240C2—81no—
241C2—122no—
242C2#563111yescharacteristic
243C2C2×#91111yescharacteristic
244C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
prime radical#5551C2
hypercenter#474D3
hypocenterC3#3353
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #5732C2semidirect
3. C3×#474C2semidirect
4. #5078C2semidirect
5. C4×#214C2semidirect
6. #4884C2semidirect
7. #4881C2semidirect
8. #4851C2semidirect
9. #4838C2semidirect
10. #4780C2semidirect
11. #4779C2semidirect
48. #474D3semidirect
140. C3⋊C8#41semidirect
239. C3#3353semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#57322
#55512
C3×#4742
#50782
C4×#2142
#48842
#48812
#48512
#48382
#47802
#47792
#33533